Different teams, contexts, expectations, resources... require different skillsets from the manager. I don't think there has ever existed a manager who would be perfectly suitable to every job. Managers who don't come with a dogmatic way of playing and whose personality is not that of a protagonists alpha tend to be better suited for top quality squads where man management and fostering a conflict free environment where big name players are allowed the freedom to flourish are the key attributes. This dynamic is more often than not suited to knockout football where the team needs to perform in one off games and find ways of winning. Real Madrid are a classic example of that, they succeed mostly with the Zidane and Ancelotti types. The same names might not perform as well if put in an enviornment where they have to build a style of playing and be the main man among relatively inexperienced players at a club that doesn't necessarily have a strong winning culture.
International teams do fall in that category most of the time, hence the success of the likes of Aimé Jaquet, Vicente del Bosque or Joachim Low. These managers had at their disposal players who already know how to play but needed to be more managed than necessarily coached. I think Southgate falls into that category which is why I find the criticism of him unfair. He is the victim of the modern trend and rise of dogmatic managers which is a relatively new thing in the Premier League. English fans are looking at Pep, Klopp and Arteta and measuring him against that type. He is not that and will never be, but more importantly, he doesn't really need to be. In that sense, he really is as good a fit as you can find for the England team whereas someone else might be a better coach, a skillset which might not have as high potential of actually making a difference in the tournament environment, but lacking in what Southgate has, a deep understanding of English football culture and the mentality of the English player and how to treat them.
